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Theorem elon 4417
Description: An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
Hypothesis
Ref Expression
elon.1  |-  A  e. 
_V
Assertion
Ref Expression
elon  |-  ( A  e.  On  <->  Ord  A )

Proof of Theorem elon
StepHypRef Expression
1 elon.1 . 2  |-  A  e. 
_V
2 elong 4416 . 2  |-  ( A  e.  _V  ->  ( A  e.  On  <->  Ord  A ) )
31, 2ax-mp 8 1  |-  ( A  e.  On  <->  Ord  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    e. wcel 1696   _Vcvv 2801   Ord word 4407   Oncon0 4408
This theorem is referenced by:  tron  4431  0elon  4461  smogt  6400  rdglim2  6461  omeulem1  6596  isfinite2  7131  r0weon  7656  cflim3  7904  inar1  8413  tfrALTlem  24347  ellimits  24521  limitssson  24522  dford3lem2  27223
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-v 2803  df-in 3172  df-ss 3179  df-uni 3844  df-tr 4130  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412
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